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Vectors and spaces

### Let $$\displaystyle{A}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{a}&{b}\backslash{c}&{d}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ and 'k' be the scalar. Find the formula that relates 'detKA' to 'K' and 'detA''

Vectors and spaces

### Let the vector space $$P^{2}$$ have the inner product $$\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.$$ Find the following for $$p = 1\ and\ q = x^{2}.$$ $$(a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)$$

Vectors and spaces

### Find all scalars $$c_{1} , c_{2}, c_{3}$$ such that $$c_{1}(1 , -1, 0) + c_{2}(4, 5, 1) + c_{3}(0, 1, 5) = (3, 2, -19)$$

Vectors and spaces

### The position vector $$\displaystyle{r}{\left({t}\right)}={\left\langle{n}{t},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}$$ describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of $$\displaystyle{t}=\sqrt{{3}}$$

Vectors and spaces

### A city planner wanted to place the new town library at site A. The mayor thought that it would be better at site B. What transformations were applied to the building at site A to relocate the building to site B? Did the mayor change the size or orientation of the library? [Pic]

Matrix transformations

### Give the elementary matrix that converts [-2,-2,-1,-3,-1,-3,1,-4,-3] to [-6,-2,-1,-5,-1,-3,-7,-4,-3]

Alternate coordinate systems

### Find the value of x or y so that the line passing through the given points has the given slope. (9, 3), (-6, 7y), $$m = 3$$

Forms of linear equations

### Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$ $$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$ $$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$

Matrix transformations

### Show that $$\displaystyle{C}^{\ast}={R}^{\ast}+\times{T},\text{where}\ {C}^{\ast}$$ is the multiplicative group of non-zero complex numbers, T is the group of complex numbers of modulus equal to 1, $$\displaystyle{R}^{\ast}+$$ is the multiplicative group of positive real numbers.

Vectors and spaces

### Use $$\displaystyle{A}{B}←→$$ and $$\displaystyle{C}{D}←→$$ to answer the question. $$\displaystyle{A}{B}←→$$ contains the points A(2,1) and B(3,4). $$\displaystyle{C}{D}←→$$ contains the points C(−2,−1) and D(1,−2). Is $$\displaystyle{A}{B}←→$$ perpendicular to $$\displaystyle{C}{D}←→$$? Why or why not?

Matrix transformations

### Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of 30◦. (b) Find the standard matrix of T , [T ]. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!

Vectors and spaces

### Let u,v1 and v2 be vectors in R^3, and let c1 and c2 be scalars. If u is orthogonal to both v1 and v2, prove that u is orthogonal to the vector c1v1+c2v2.

Vectors and spaces

### Let U,V be subspaces of Rn. Suppose that U⊥V. Prove that {u,v} is linearly independent for any nonzero vectors u∈U,v∈V.

Alternate coordinate systems

### To plot: Thepoints which has polar coordinate $$\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}$$ also two alternaitve sets for the same.

Vectors and spaces

### Write an equation of the line that passes through (3, 1) and (0, 10)

Alternate coordinate systems

### The system of equation $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{b}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{g}\right\rbrace}\in{\left\lbrace\le{f}{t}{\left\lbrace{\left\lbrace{c}\right\rbrace}{\left\lbrace{a}\right\rbrace}{\left\lbrace{s}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{s}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right\rbrace}{\frac{{{n}}}{{{\left\lbrace{2}\right\rbrace}}}}\backslash-{\frac{{{\left\lbrace{\left\lbrace{n}\right\rbrace}+{\left\lbrace{1}\right\rbrace}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}{\left\lbrace{n}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace\le{f}{t}{\left\lbrace{\left\lbrace{c}\right\rbrace}{\left\lbrace{a}\right\rbrace}{\left\lbrace{s}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{s}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right\rbrace}$$ by graphing method and if the system has no solution then the solution is inconsistent. Given: The linear equations is $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{2}\right\rbrace}{\left\lbrace{x}\right\rbrace}+{\left\lbrace{y}\right\rbrace}={\left\lbrace{1}\right\rbrace}{\left\lbrace\quad\text{and}\quad\right\rbrace}{\left\lbrace{4}\right\rbrace}{\left\lbrace{x}\right\rbrace}+{\left\lbrace{2}\right\rbrace}{\left\lbrace{y}\right\rbrace}={\left\lbrace{3}\right\rbrace}.$$

Alternate coordinate systems

### What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?

Alternate coordinate systems

### The equivalent polar equation for the given rectangular - coordinate equation. $$\displaystyle{y}=\ -{3}$$

Alternate coordinate systems